1 Answer
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You can use Stirling's approximation. Considering the multinomial coefficient in $a_i$ instead of what you wrote, and putting $p_i = a_i/\sum_i a_i$, you should get that the logarithm of the multinomial coefficient is
$$(1+o(1))(\sum_i a_i) H(p_1,\ldots,p_n),$$
where $H(\cdots)$ is the entropy function:
$$H(p_1,\ldots,p_n) = -\sum_i p_i \log p_i.$$